Coupled Membranes with Doubly Negative Mass Density and Bulk Modulus

Min Yang,1 Guancong Ma,1 Zhiyu Yang,1 and Ping Sheng1,2

1Department of Physics, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, China2Institute for Advanced Study, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, China

(Received 9 October 2012; published 28 March 2013)

We present a structurally and conceptually simple acoustic double negative metamaterial comprising

two coupled membranes. Owing to its symmetry, the system can generate both monopolar and dipolar

resonances that are separately tunable, thereby making broadband double negativity possible. A homo-

genization scheme is implemented that enables the exact characterization of our metamaterial by

the effective mass density and bulk modulus even beyond the usual long-wavelength regime, with the

measured displacement fields on the sample’s surfaces as inputs. Double negativity is achieved in the

frequency range of 520–830 Hz. Transmission and reflection predictions using effective parameters

are shown to agree remarkably well with the experiment.

DOI: 10.1103/PhysRevLett.110.134301 PACS numbers: 43.20.+g, 43.40.+s, 46.40.�f

The emergence of acoustic and elastic metamaterials hassignificantly broadened the horizon for acoustic or elasticwaves. Novel phenomena such as focusing and subdiffrac-tion imagining [1–4], near-field amplification [5], cloaking[6–9], localization of ultrasound [10], one-way transmis-sion [11–13], and super absorption [14] have been pro-posed or experimentally demonstrated. At the core ofthese phenomena are the resonance-induced effective ma-terial characteristics such as the negative effective mass[5,15–18], negative effective modulus [19,20], or negativeshear modulus [21,22].

With the realization of negative dynamic mass density inthe membrane system [16] and negative modulus via theHelmholtz resonators (HR) [19], a natural extension is thedoubly negative acoustic metamaterial based on the com-bination of the two [23]. However, coupling between themembrane and the HR is not straightforward, as the geo-metric symmetries of the two systems are different. Henceit is only recently that such an attempt was successfullydemonstrated in the low frequency limit [24,25]. However,combining two different structures is not the only strategyto the realization of acoustic double negativity. It wasshown theoretically that a single resonating unit can dis-play overlapping responses in multiple angular momentumchannels [21], including monopolar and dipolar, which arethe key to the realization of acoustic double negativity [26].Other routes to achieve double negativity, such as thegeometric-induced band folding [27], have also been dem-onstrated theoretically and numerically.

In this Letter, we show both experimentally and theo-retically the realization of double negativity in a singleresonating structure, based on two coupled membranes.The work was motivated primarily by symmetry consid-erations. Consider a homogeneous 1D system of finiteextent. The (transversely averaged) motions of two endsurfaces represent two independent degrees of freedom—asymmetric mode in which the two surfaces move in phase,

and an antisymmetric mode in which the two surfacesmove out of phase with each other. These two modes cancharacterize, respectively, the dipolar and monopolar reso-nances of the system. They correspond to the two effectiveparameters of the system because in the dipolar case thereis center of mass motion and hence the correspondingresonance is denoted masslike, whereas in the monopolarcase the center of mass remains stationary and only thebulk modulus plays a role; hence, it is denoted bulk mod-uluslike [26]. Provided the eigenfrequencies of the tworesonances are tunable via system parameters, it is straight-forward to realize double negativity in a desired frequencyrange. A distinct advantage of such a system is that, dueto the simplicity and symmetry of our metamaterial, wecan afford exact characterization of the system by usingthe measured end surface displacements, even beyond theusual long-wavelength regime [21,28,29]. From theextracted effective parameter values, the predicted trans-mission and reflection spectra can be compared to theexperiment. In what follows we show that such a simple1D system can indeed be realized, by coupling two mem-branes. While the membranes would inevitably have trans-verse dimensions, it is shown that in the low frequencylimit where the relevant wavelength is much larger than thetransverse dimension, the system is accurately 1D as far asthe propagating waves are concerned.In Fig. 1(a) we show the structure of the metamaterial

that comprises two identical circular membranes, eachwith radius R ¼ 14 mm, thickness 0.2 mm, and decoratedby a circular rigid platelet (radius of 4.5 mm and mass of159 mg) attached to the center. The two membranes areeach fixed to a rigid cylindrical side wall with a radialtensile stress 1:3� 106 Pa. They are connected by anacrylic plastic ring, which has a thickness of 1.5, innerradius 10, and height 6.0 mm. The ring has a mass of395 mg, and the materials parameters of the membranescan be found in Ref. [14]. The amplitude and phase of the

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transmission and reflection were measured in a modifiedimpedance tube apparatus, comprising two Bruel and Kjærtype-4206 impedance tubes with the sample sandwiched inbetween [30]. The front tube has two sensors, plus a loud-speaker at one end to generate a plane wave in the tube.The back tube has one sensor to measure the transmittedwave. Figures 2(a) and 2(b) show the measured transmis-sion and reflection amplitudes, respectively. Three trans-mission peaks, located at 290.1, 522.6, and 834.1 Hzare seen.

We limit the relevant acoustic frequency f by the con-dition v0=f ¼ � > 2R, where v0 ¼ 343 m= sec is thespeed of sound in air. Thus we have f < 1:225� 104 Hzunder this constraint. An immediate consequence is that asfar as the radiation modes are concerned, i.e., transmissionand reflection, the system may be accurately considered asone dimensional. This can be seen as follows. The normaldisplacement u of the membrane may be decomposed asu ¼ hui þ �u, where hui represents the surface-averagednormal displacement of the membrane (here hi representssurface averaging) and �u the fine details of the membranemotion. In the air layer next to the membrane surface, theacoustic wave must satisfy the dispersion relation k2k þk2? ¼ ð2�=�Þ2, where kkð?Þ represents the wave vector

component parallel (perpendicular) to the membrane sur-face. Because the two-dimensional fine pattern of kk can bedescribed by a linear superposition of kk’s, all of whichmust be greater than 2�=2R> 2�=�, it follows that the

relevant k2? < 0. That is, the displacement component �u

leads only to evanescent, nonradiating modes. The dis-placement component, hui, on the other hand, representsthe pistonlike motion of the membrane and has kkcomponents peaked at kk ¼ 0; hence, it is coupled to the

radiation modes.Simplification to a 1D system greatly facilitates the

visualization of the relevant symmetries of the two typesof resonances, involving either the in-phase or the out-of-phase motion of the two membranes. An important elementof our experimental measurements is the use of laservibrometer (Graphtec AT500-05) to map the normal dis-placement u across the membrane on the transmission side,plus the relative phase �� between the two membranes,which can be detected by the relative motion between thetwo platelets. In Figs. 1(b)–1(d) we show three displace-ment fields of the coupled-membrane system at the trans-mission peaks, i.e., resonance frequencies (f1þ ¼ 290:1,

f2� ¼ 522:6, and f3þ ¼ 834:1 Hz). The black curves

delineate the simulated results by using the COMSOL multi-physics finite element package, whereas the red circlesrepresent the measured results using laser vibrometer.Excellent agreement is seen. For the first mode, bothmembranes oscillate in unison, carrying the ring togetherin a translational motion. For the second eigenmode, thering is motionless and only the membranes vibrate.Because the acrylic plastic ring is rather rigid, it is impos-sible for the soft membrane to compress the ring at suchlow frequencies. Consequently, the ring acts like an anchor,and the central portions of the two membranes vibrate in anout-of-phase manner. For the third eigenmode, the ring and

FIG. 2 (color online). (a) Transmission coefficient and(b) reflection coefficient of the metamaterial. The open circlesare the experimental results obtained using the impedance mea-surements; the solid curves are calculated from �� and �� shownin Fig. 3. Excellent agreement is seen. The band gaps are shadedgray, the double-negative passband (between 520 and 830 Hz) islight gray (yellow), and the double-positive passband is withoutshading.

FIG. 1 (color online). (a) Schematic drawing of the metama-terial. (b) Numerically simulated (black curves) and experimen-tally measured (red circles) vibration profiles (eigenfunctions) ofthe first eigenmode. Here and in (c) and (d) circular symmetry isassumed. The rigid circular platelet is shown schematically bythe hollow rectangle. (c) The second eigenmode. (d) The thirdeigenmode. For the third eigenmode, the relative phase is notedto be somewhat larger than that for the first mode. Here f� isthe corresponding eigenfrequency and ��� the relative phasedifference between the two central plates.

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the platelets vibrate in opposite phase, but the motion of thetwo membranes is in unison. It is seen that the simulatedphase relation between the two platelets agrees with theexperimental results almost perfectly.

While the first and third eigenmodes are clearly dipolarin character and hence of mass-density type (MDT), thesecond mode has the monopolar symmetry and hence is ofbulk-modulus type (BMT) [26]. For the dipolar resonance,the total mass of the ring and the platelets serves asthe most important parameter for tuning its frequency.For the monopolar resonance, the membranes’ separationand transverse dimension are the crucial parameters. Thefourth eigenmode is noted to be at a much higher frequencyof 2976.3 Hz. Its effect in the frequency range of ourinterest is minimal, and therefore it is ignored in thefollowing analysis.

The two relevant effective material parameters are thedynamic mass density �� (arising from the symmetricmode) and the effective bulk modulus �� (arising fromthe antisymmetric mode). To extract these two effectiveparameters, we propose a homogenization scheme, withdetailed derivation presented in the Supplemental Material[31], based on the fact that the behavior of our system isdictated by the resonant eigenmodes. The scheme needsonly the three relevant eigenfunctions to delineate thecorrelated motions on the two ends (i.e., the two mem-branes). This aspect is distinct from the homogenizationschemes in which matching the response of the entirefrequency range of interest is required [21,32–34].

Consider the eigenfunction expansion of the scalarGreen function [29],

Gð ~x; ~x0Þ ¼ X

�

u��ð ~xÞu�ð ~x0Þ��ð!2

� þ i!�� �!2Þ ; (1)

where angular frequency ! ¼ 2�f, ���R�u

��ð ~xÞ�ð ~xÞ�

u�ð ~xÞd~x denotes the averaged mass density for the �theigenfunction u�ð ~xÞ, �ð ~xÞ is the local mass density, and!�

and �� are the corresponding resonant frequency anddissipation coefficient. By using the experimentally mea-sured eigenfunctions u�ð ~xÞ [as shown in Figs. 1(b)–1(d), inwhich circular symmetry is assumed], the relevant �� canbe evaluated from its definition. The dissipation coeffi-cients ��’s can be evaluated from the experimentallymeasured membrane displacement, with details shown inthe Supplemental Material [31]. For the frequency rangeof interest, it turns out that only three eigenfunctionsare needed; i.e., � ranges from 1 to 3 only. In Table Iwe present the relevant parameter values of the threeeigenfunctions, evaluated from the measured membranedisplacement as shown in Fig. 1.

By carrying out the cross-sectional average on G(denoted by hGi), we obtain a symmetric component, �Gþ,and an antisymmetric component, �G�, where

�G� ¼ hGðx0; x0Þi � hGðx0;�x0Þi

¼ X3

�¼1

hu��ðx0Þi½hu�ðx0Þi � hu�ð�x0Þi���ð!2

� þ i!�� �!2Þ ; (2)

and the two coordinates are now specified at the positionsof the two coupled membranes. The cross-sectional aver-aged values of hu�ð�x0Þi can be directly obtained from thelaser vibrometer data (Fig. 1) for the three eigenfunctions;hence, together with the values of �� and �� (Table I) wecan obtain the numerical values of �G� for all the relevantfrequencies.Now consider a homogeneous 1D system of length 2x0.

The Green function of such a 1D system is uniquelydetermined by the two material parameters �� and ��. In

particular, we can have the similar quantities �Gð1DÞ� that are

given by the formulas [35]

�Gð1DÞþ ¼ � cotðx0! ffiffiffiffi

��p

=ffiffiffiffi��

p Þ!

ffiffiffiffi��

p ffiffiffiffi��

p ; (3a)

�Gð1DÞ� ¼ tanðx0! ffiffiffiffi��

p=

ffiffiffiffi��

p Þ!

ffiffiffiffi��

p ffiffiffiffi��

p : (3b)

By requiring �G� ¼ �Gð1DÞ� , we obtain two sets of equations

that determine �� and �� as functions of frequency. As thesolutions are multiple valued, we select the branch withthe longest wavelength. Results are shown in Figs. 3(a)and 3(b). For the purpose of clarity, only the real parts ofthe effective parameters are plotted. The correspondingimaginary parts are shown in the Supplemental Material[31]. The frequency range in light gray (yellow) denotesthe double negativity regime—520 to 830 Hz—which isnoted to be fairly broad.It should be noted that in the above homogenization

scheme no use has been made of the usual long-wavelengthapproximation. Hence, even if the effective wavelengthin our sample is comparable to 2x0, the relevant effectiveparameter values should still be accurate as far as thesurface response (and hence the far-field wave field) ofthe system is concerned.From �� and ��, we can analytically calculate the trans-

mission and reflection coefficients T and R from the 1Dmodel (see the Supplemental Material [31]). The resultsare displayed in Fig. 2 as solid curves. They are seento agree remarkably well with the experiments (opencircles) with no adjustable parameters, even beyond the

TABLE I. Values of f�, ��, and �� for the first three eigen-functions.

� 1 2 3

f� (Hz) 290.1 522.6 834.1

�� (104 kg=m3) 2.72 2.95 1.19

�� (Hz) 10.20 12.79 49.94

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usual long-wavelength regime, e.g., around f~1þ and f2�where the half wavelength is on the order of 2x0.

By periodically repeating the unit cells in the planeparallel to the membranes, a planar array [36–38] or awall with overall transport properties similar to a singleunit cell can be constructed. In Sec. III of the SupplementalMaterial [31] we give an experimental demonstrationshowing that the transmission and reflection characteristicsof an array, comprising identically constructed units, issimilar to a single cell. It confirms the fact that each cellis well isolated from its neighbors.

The transmission properties of our metamaterial aredetermined by two factors: impedance matching with airand the values of effective wave vectors. We notice thatReð ��Þ crosses zero precisely at the two eigenfrequenciesf1þ and f3þ [Fig. 3(a)], arising from the two dipolar

resonances. A direct consequence is that the effectiveimpedance j �Zj ¼ j ffiffiffiffiffiffiffiffi

�� ��p j [Fig. 3(d)] matches well with

the background air. Two transmission peaks, accompaniedby reflection minima (Fig. 2), are seen at f1þ and f3þ .

The antiresonance frequency, represented by f~1þ , is

located between the two MDT eigenfrequencies due tothe out-of-phase hybridization of the two neighboringMDT eigenmodes that implies huðx0Þi þ huð�x0Þi ¼ 0[16]. To simplify the picture, we have tuned the BMTfrequency f2� to coincide with the antiresonance f~1þ to

within several Hz. Owing to dissipative broadening, ourcalculation shows j �Zj to display a small dip but the value isstill fairly large in this frequency range [Fig. 3(d)] andhence mismatches with air. The transmission dip, usuallyexpected at the antiresonance frequency f~1þ , is effectively

countered by the BMT resonance. However, the trans-mission peak at this frequency [Fig. 3(c)] is due to theFabry-Perot effect of constructive interference. That is,

�k ¼ �=2x0 at f2� , with 2x0 being the thickness of the

metamaterial.The key frequencies discussed above, f1þ , f~1þ (f2�), and

f3þ , divide the spectrum into two passbands. The first one

[white regions in Figs. 2 and 3 with f 2 ðf1þ ; f~1þðf2�ÞÞ], isa conventional double-positive band. The second one[light gray (yellow) regions in Figs. 2 and 3 with f2ðf~1þðf2�Þ;f3þÞ], is due to the overlapping of the negative

�� and negative �� bands. In the doubly negative frequency

regime, the effective wave vector �k ¼ !ffiffiffiffiffiffiffiffiffi��= ��

pis real;

hence, the acoustic wave is propagative. However, thewave response is out of phase to that of the double-positivemedium, as can be seen from the negative slope of thedispersion in Fig. 3(c).Single-negative band gaps are found in two regimes,

f < f1þ and f > f3þ (the gray bands in Figs. 2 and 3).

The first gap is due to negative-valued ��, whereas thesecond gap is due to the negative ��. Single negativity inthe effective parameters gives rise to pronounced imaginarypart of the effective wave vectors within the band gaps evenin the absence of dissipative effects, so that the acousticwave must be evanescent. However, we notice that thetransmission coefficients within the band gaps are notnecessarily small. This is due to the relatively long decaylength, given by d ¼ Imð �kÞ�1. The minimum d is�13 mm[31], which is still larger than the thickness of our sample.Hence the evanescent soundwave is penetrative in this case.To conclude, we have implemented a very simple acous-

tic metamaterial using two coupled membranes in whichthe effective parameters can be exactly characterized.Double negativity in both the dynamic mass density andeffective bulk modulus is achieved in a fairly broad fre-quency regime. The simplicity of the structure makes themsuitable as the basic units for constructing acoustic

FIG. 3 (color online). Effective parameters of the metamaterial. For clarity, only the real parts of the dynamic mass density �� (a) andeffective bulk modulus �� (b) are shown. These effective parameters are, respectively, normalized to the static mass density of lead,�lead ¼ 1:13� 104 kg=m3, and the typical static Young’s modulus of latex rubber, Erubber ¼ 1:9� 106 Pa. The real part of theeffective wave vector �k and the magnitude of the effective impedance �Z are shown in (c) and (d), respectively, with �Z normalized by~Z ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

�leadErubber

p ¼ 1:5� 105 N sm�3. The band gaps are shaded gray, the double-negative passband is light gray (yellow), and thedouble-positive passband is white.

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metamaterial devices with complex functionalities.Examples include but are not limited to acoustic super-lensing, acoustic Fresnel zone plates, gradient-indexdevices, and even near-field devices. In addition, the pro-posed theory framework makes extraction of effectiveparameters relatively simple and accurate, thereby allow-ing for easier and more precise control over the deviceproperties.

This work is supported by Hong Kong RGC GrantsNo. HKUST 604207, No. HKUST 606611, andNo. HKUST2/CRF/11G. M.Y. and G.M. contributedequally to this work.

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1

Supplementary Information on

“Coupled Membranes with Doubly Negative Mass Density and Bulk Modulus”

by

Min Yang , Guancong Ma, Zhiyu Yang and Ping Sheng

I. Exact Effective Parameters Characterization Using Surface Response

The purpose of a homogenization scheme is to extract a small number of effective

parameters for the efficient description of composite properties as perceived by an external

observer. In the present case, we employ the motions at the two bounding surfaces as the central

element in our approach.

From either numerical simulation (solid curves in Fig. 1) or experimental measurements,

such as those by laser vibrometer (open circles in Fig. 1), the eigenstates ( )u x of the

composite material form an orthonormal complete set for any of its possible motion field ( )u x .

Here u denotes the normal displacement of the 1D system along the thk direction. For each

eigenstate, a generalized mass density is defined as

* ( ) ( ) ( )u x x u x dx

, (S1a)

resonance angular frequency 2 f is given by

2 * ( ) ( ) ( ) /ikkm

i m

u x C x u x dxx x

, (S1b)

where summation over indices i and m is assumed; and generalized dissipation coefficient (for

) is given by [1]

* ( ) ( ) ( ) /ikkm

i m

u x x u x dxx x

. (S1c)

Here ( )x is the spatial distributed mass density for the heterogeneous composite, and

2

( )ikkmC x and ( )ikkm x are the 4 th -ranked tensors for its elastic modulus and dissipation

coefficients, respectively, specialized to the case when all the displacements are along the thk

direction. These quantities provide a formal eigenfunction expansion for the Green function

[2-4]:

*

2 2

( ) ( )( , )

( )

u x u xG x x

i

. (S2)

It is seen from Eq. (S2) that each term in the summation is dominant in the frequency regime

around its eigenfrequency . Hence, for the purpose of investigation in our finite (and low)

frequency range, only a few eigenstates are needed.

As argued in the text, as long as we are interested only in the propagating radiation modes,

the problem is effectively one-dimensional (1D). For a 1D system, the external wave field is

only sensitive to the motion of the two bounding surfaces. By averaging over the cross section,

a set of 1D surface response functions can be obtained as

*

0 0 00 0 0 0 2 2

( ) ( ( ) ( ) )( , ) ( ),

( )

u x u x u xG G x x G x

ix

, (S3)

where the symbol stands for surface averaging, and 0x denote the positions of the two

bounding surfaces.

Now consider a homogeneous 1D sample with length 2x0. The eigenfunctions of such a

system are simply given by

0

0 0

( )1( ) cos , 1,2,3

2

x xu x

x x

with0 0( ) 1/ 2u x x . The relevant eigenfrequencies are given by 0/ / (2 )x . Here

and are the bulk modulus and mass density of the homogeneous system. If we require the

external wave response of this homogeneous 1D system to be identical to that of our

experimental system, it is obvious that such a requirement can be accurately satisfied since the

3

external wave field is sensitive only to the motion of the two bounding surfaces, and we have

exactly two effective parameters, and ,for satisfying this constraint at each frequency (this

also makes clear why our scheme works in 1D, but in higher dimensions the long-wavelength

approximation becomes necessary). It turns out that for the 1D homogeneous systems, the

eigenfunction expansion of the Green function can be analytically evaluated, with the results

shown in the text, reproduced here for completeness:

(1D) 0cot( / )xG

, (S4a)

(1D) 0tan( / )xG

. (S4b)

The homogenization condition is then simply

(1D)GG . (S5)

Substitution of Eq. (S4) into Eq. (S5) leads to analytical solutions for the effective dynamic

mass density and the effective bulk modulus as

2

0

arctan( / )( )

G G

x G G

, (S6a)

0( )arctan( / )

x

G G G G

. (S6b)

Here the values of G at each frequency can be directly determined from experimentally

measured surface displacement plus the values of the (which can be easily evaluated) and

(see below).

The effective wave-vector k and effective impedance Z can also be evaluated as

0

( ) 1( ) arctan /

( )k G G

x

, (S7a)

1( ) ( ) ( )Z

G G

. (S7b)

4

The results of these evaluations are shown in Fig. S1.

II. Evaluation of Dissipation Coefficients αβ and Mass Density αρ

Even with its definition in Eq. (S1c), the dissipation coefficients are not easy to be

determined directly since the dissipative tensor ikkm is usually not available. However, these

dissipation coefficients can still be extracted from the magnitude of surface motion during wave

scattering, which can be measured by laser vibrometer.

At resonance, the system is strongly coupled to either monopolar or dipolar motions, so that

its surface normal displacement average at 0x may be written as

0 0 0 0( ) ( ) (1 )2

iu x F x G Z u S G , (S8)

where 0( )F x denotes the force acting on the material from air at 0x , 0Z is the impedance of air,

0u is the maximum displacement of the incident wave, and 0 0/S Z G i Z G i are

the scattering coefficients for the system for the symmetric and anti-symmetric modes (indicated

by the subscript sign) [5]. Here u denotes the displacement to be that measured

experimentally, instead of the normalized eigenfunction u .

We note that at resonance, G is related to its eigenfunction 0( )u x as

2

02 | ( ) |u xG

i

. (S9)

Substitution of (S9) into (S8) leads to

2

0 0

0 02

0 0

2 ( ) | /( )

2 ( ) |

|

/|

Z u xu x u

Z u x

. (S10)

Therefore, the dissipation coefficient may be directly solved from Eq. (S10) as

2

0 0 0

0

2 ( ) |1

(

|

)

Z u x u

u x

. (S11)

More explicitly, if we relate u and u by the relation

5

0 0, ,( ) ( ) /u x r u x r A , (S12)

then A is given by

2 2[ ]A u x dx

p2 2

rubber plate 00 0

4 ( ) ( ),R r

h u r rdr h u x r rdr

r2

r1

2

ring 0 ,2 ( )r

rh u x r rdr . (S13)

The three separate integrals arise from the three separate regions that can be clearly visualized in

Fig. 1. From Eq. (S11), we then obtain

22

0 002 0

00

(8

, )( , )

12

R

R

Z ux r rdr

R x r rdr

Ru

u

. (S14)

For the mass density , we can evaluate its value directly as

2

rubber 0 rubber0

2 2 ( ),R

h u x r rdr

p 2

plate 0 plate0

2 ( ,2 )r

h u x r rdr

r2

r1

2

ring 0 ring,2 ( )r

rh u x r rdr . (S15)

The evaluated and for the three relevant eigenfunctions are presented in Table I.

In the following, we show in Fig. S1 the evaluated dynamic mass density, effective bulk

modulus, the effective wave vector, and the impedance of our sample, all as functions of

frequency based on both numerically simulated eigenfunctions (open triangles) and

experimentally measured eigenfunctions (solid curves). The imaginary parts are shown in red,

the real parts in black. In Fig. S2 we show the transmission and reflection predictions obtained

from the effective parameters in the homogeneous 1D model. Excellent agreement is seen.

6

III. Transmission Through a Planar Array

As mentioned in the text, the locally resonant nature of the coupled-membrane system

enables it to be repeated and fit onto an existing wall / skeleton, forming a planar array without

changing its characteristics, as long as each cell is well isolated from its neighbors. A wall

comprising a large array of such unit cells can exhibit dramatic frequency dependent behavior,

predictable from the effective parameters of the unit cell.

To demonstrate this, we manufactured a set of coupled-membranes systems with radius of 12

mm, the two stiff plates have radius of 4.5 mm and weight of 190 mg. In the middle we place a

10 mm height rigid ring that has a mass of 380 mg. The ring has an outer radius of 10.5 mm and

thickness of 1.5 mm. Note that due to the constraints imposed by our experimental setup and

fabrication ability, the unit cell’s parameters are slightly different from those in the main text. By

arranging five such unit cells them into a 50 mm radius circular aluminum plate (Fig. S3(a)), the

intensity transmission coefficient has been measured under normal incidence (open circles in Fig.

S3(b)). The measured result for a single cell is also plotted in Fig. S3(b) as red symbols.

Based on the homogenized effective parameters, the amplitude transmission coefficient for the

metamaterial-modified plate can be easily calculated. The results for the two cases are shown as

solid curves in Fig. S3(b). Excellent agreement is seen. The similarity between the array and

single cell is also each to see.

7

FIGURE S1. Effective parameters of the metamaterial. The dynamic mass density (a) and

effective bulk modulus (b) are respectively normalized to the static mass density of lead,

4 3

lead 1.13 10 kg/m , and the typical static Young's modulus of latex rubber, 6

rubber 1.9 10E Pa.

The wave-vector k and the effective impedance Z are shown in (c) and (d), respectively, with Z

normalized by 5 -3

lead ruuber 1.5 10 N s mZ E . The solid curves are results based on measured

eigenfunctions and the open triangles are from numerically simulated eigenfunctions. Real parts of the

parameters are colored in black, and imaginary parts are in red. The bandgaps are shaded in grey, the

double-negative passband is colored in yellow, and the double-positive passband is in white.

8

FIGURE S2. (a) Transmission coefficient and (b) reflection coefficients of the metamaterial. The open

circles are the experimental results obtained using the impedance measurements, the solid curves are

calculated from and shown in Fig. S1. Excellent agreement is seen. Color black is standing for

amplitudes and color red is for phases. The bandgaps are shaded in gray, the double-negative passband is

shaded in yellow, and the double-positive passband is without shading.

9

FIGURE S3. (a) A photo image of an array comprising five unit cells. The cyan circle represents the

cross section of our impedance tube. (b) Normalized transmission plotted as a function of frequency f . The

open circles are the experimental results obtained using the impedance measurements, the solid curves

are calculated from the homogenized effective parameters. Black stands for results of five-unit planar

array and red represents the transmission for a single unit cell. The bandgaps are shaded in gray, the

double-negative passband is shaded in yellow, and the double-positive passband is without shading.

Except for the magnitude, the similarity of the two curves is clearly seen, thereby confirming the good

isolation between the unit cells.

10

References

[1] L. Landau and E. Lifshitz, Theory of elasticity," (Pergamon Press, 1989) Chap. 5, p. 135.

[2] J. Mathews and R. Walker, Methods of mathematical physics," (Addison-Wesley, 1970)

Chap. 9, p. 267.

[3] Courant and Hilbert, Methods of mathematical physics 2 vols," (John Wiley & Sons, 1989)

Chap. 5, p. 360.

[4] J. Willis, in Continuum Micromechanics, edited by P. Suquet, CISM Courses and Lectures

No. 377 (Springer Wien, New York, 1997).

[5] D. Keefe, R. Ling, and J. Bulen, The Journal of the Acoustical Society of America 91, 470

(1992).